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Maximal $ L^p$ analysis of finite element solutions for parabolic equations with nonsmooth coefficients in convex polyhedra


Authors: Buyang Li and Weiwei Sun
Journal: Math. Comp. 86 (2017), 1071-1102
MSC (2010): Primary 65M12, 65M60; Secondary 35K20
DOI: https://doi.org/10.1090/mcom/3133
Published electronically: August 18, 2016
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Abstract: The paper is concerned with Galerkin finite element solutions of parabolic equations in a convex polygon or polyhedron with a diffusion coefficient in $ W^{1,N+\alpha }$ for some $ \alpha >0$, where $ N$ denotes the dimension of the domain. We prove the analyticity of the semigroup generated by the discrete elliptic operator, the discrete maximal $ L^p$ regularity and the optimal $ L^p$ error estimate of the finite element solution for the parabolic equation.


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Additional Information

Buyang Li
Affiliation: Mathematisches Institut, Universität Tübingen, D-72076 Tübingen, Germany – and – Department of Mathematics, Nanjing University, Nanjing 210093, People’s Republic of China
Address at time of publication: Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong
Email: buyang.li@polyu.edu.hk

Weiwei Sun
Affiliation: Department of Mathematics, City University of Hong Kong, Hong Kong
Email: maweiw@math.cityu.edu.hk

DOI: https://doi.org/10.1090/mcom/3133
Received by editor(s): January 13, 2015
Received by editor(s) in revised form: October 19, 2015
Published electronically: August 18, 2016
Additional Notes: The work of the first author was supported in part by NSFC (grant no. 11301262), and the research stay of the author at Universität Tübingen was supported by the Alexander von Humboldt Foundation
The work of the second author was supported in part by a grant from the Research Grants Council of the Hong Kong SAR, China (project no. CityU 11302915)
Article copyright: © Copyright 2016 American Mathematical Society